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Duality Based Adaptivity Of Model And Discretization In Multiscale Finite Element Methods
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Book Synopsis Duality-based Adaptivity of Model and Discretization in Multiscale Finite-element Methods by : Matthias Sebastian Maier
Download or read book Duality-based Adaptivity of Model and Discretization in Multiscale Finite-element Methods written by Matthias Sebastian Maier and published by . This book was released on 2015 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Finite Element Methods for Differential Equations by : Wolfgang Bangerth
Download or read book Adaptive Finite Element Methods for Differential Equations written by Wolfgang Bangerth and published by Birkhäuser. This book was released on 2013-11-11 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.
Book Synopsis Duality-based Adaptivity in the Hp-finite Element Method by : Vincent Heuveline
Download or read book Duality-based Adaptivity in the Hp-finite Element Method written by Vincent Heuveline and published by . This book was released on 2003 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Finite Elements in the Discretization of Parabolic Problems by : Christian A. Möller
Download or read book Adaptive Finite Elements in the Discretization of Parabolic Problems written by Christian A. Möller and published by Logos Verlag Berlin GmbH. This book was released on 2011 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
Book Synopsis Multiscale Model Reduction by : Eric Chung
Download or read book Multiscale Model Reduction written by Eric Chung and published by Springer Nature. This book was released on 2023-06-07 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.
Book Synopsis Adaptive Finite Element Methods for Multiscale Partial Differential Equations by : Achim Nonnenmacher
Download or read book Adaptive Finite Element Methods for Multiscale Partial Differential Equations written by Achim Nonnenmacher and published by . This book was released on 2011 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Multiscale and Adaptivity: Modeling, Numerics and Applications by : Silvia Bertoluzza
Download or read book Multiscale and Adaptivity: Modeling, Numerics and Applications written by Silvia Bertoluzza and published by Springer Science & Business Media. This book was released on 2012-01-07 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of lecture notes for the CIME course on "Multiscale and Adaptivity: Modeling, Numerics and Applications," held in Cetraro (Italy), in July 2009. Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.
Book Synopsis Numerical Mathematics and Advanced Applications ENUMATH 2017 by : Florin Adrian Radu
Download or read book Numerical Mathematics and Advanced Applications ENUMATH 2017 written by Florin Adrian Radu and published by Springer. This book was released on 2019-01-05 with total page 1070 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects many of the presented papers, as plenary presentations, mini-symposia invited presentations, or contributed talks, from the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) 2017. The conference was organized by the University of Bergen, Norway from September 25 to 29, 2017. Leading experts in the field presented the latest results and ideas in the designing, implementation, and analysis of numerical algorithms as well as their applications to relevant, societal problems. ENUMATH is a series of conferences held every two years to provide a forum for discussing basic aspects and new trends in numerical mathematics and scientific and industrial applications. These discussions are upheld at the highest level of international expertise. The first ENUMATH conference was held in Paris in 1995 with successive conferences being held at various locations across Europe, including Heidelberg (1997), Jyvaskyla (1999), lschia Porto (2001), Prague (2003), Santiago de Compostela (2005), Graz (2007), Uppsala (2009), Leicester (2011), Lausanne (2013), and Ankara (2015).
Book Synopsis Adaptive Finite Element Methods for Optimization in Partial Differential Equations by :
Download or read book Adaptive Finite Element Methods for Optimization in Partial Differential Equations written by and published by . This book was released on 2000 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by (elliptic) partial differential equations. The Lagrangian formalism yields the first-order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residual-based a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and co-state variables can be used as sensitivity factors multiplying the local cell-residuals in the error estimators. This results in a generic and efficient algorithm for mesh adaptation within the optimization process. Applications of the developed method are boundary control problem models governed by Ginzburg-Landau equations (superconductivity in semi-conductors), by Navier-Stokes equations, and by the Boussinesq viscosity model (flow with temperature transport for zero gravitation). Computations with more than 2 million unknowns were performed.
Book Synopsis A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems by : Raymond Millward
Download or read book A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems written by Raymond Millward and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.
Book Synopsis A Parallel Goal-oriented Adaptive Finite Element Method for 2.5D Electromagnetic Modeling by : Kerry Key
Download or read book A Parallel Goal-oriented Adaptive Finite Element Method for 2.5D Electromagnetic Modeling written by Kerry Key and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a parallel goal-oriented adaptive finite element algorithm that can be used to rapidly compute highly accurate solutions for 2.5D controlled-source electromagnetic (CSEM) and 2D magnetotelluric (MT) modeling problems. We employ unstructured triangular grids to permit efficient discretization of complex modeling domains such as those containing topography, dipping layers and multiple scale structures. Iterative mesh refinement is guided by a goal-oriented error estimator based on a form of dual residual weighting, which is carried out using hierarchical basis computations. Our formulation of the error estimator considers the relative error in the strike aligned fields and their spatial gradients, and therefore results in a more efficient use of mesh vertices than previous error estimators based on absolute field errors. This algorithm is parallelized over frequencies, transmitters, receivers and wave-numbers, where adaptive refinement can be performed in parallel on subsets of these parameters while nearby parameters are able to share the refined grid, thus enabling our algorithm to achieve accurate solutions in run-times of seconds to tens of seconds for realistic models and data parameters when run on cluster computers containing about a thousand processors. Application of this new algorithm to a complex model that includes strong seafloor topography variations and multiple thin stacked reservoirs demonstrates the performance and scalability on a large cluster computer.
Book Synopsis A High-order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations by : Todd A. Oliver
Download or read book A High-order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations written by Todd A. Oliver and published by . This book was released on 2008 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents high-order, discontinuous Galerkin (DG) discretizations of the Reynolds-Averaged Navier-Stokes (RANS) equations and an output-based error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the Spalart-Allmaras (SA) turbulence model are examined. The dual consistency of multiple DG discretizations of the RANS-SA system is analyzed. The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent. A dual consistency correction for this discretization is derived. The analysis also demonstrates that discretizations based on the popular mixed formulation, where dependence on the state gradient is handled by introducing additional state variables, are generally asymptotically dual consistent. Numerical results are presented to confirm the results of the analysis. The output error estimation and output-based adaptation algorithms used here are extensions of methods previously developed in the finite volume and finite element communities. In particular, the methods are extended for application on the curved, highly anisotropic meshes required for boundary conforming, high-order RANS simulations. Two methods for generating such curved meshes are demonstrated. One relies on a user-defined global mapping of the physical domain to a straight meshing domain. The other uses a linear elasticity node movement scheme to add curvature to an initially linear mesh. Finally, to improve the robustness of the adaptation process, an "unsteady" algorithm, where the mesh is adapted at each time step, is presented. The goal of the unsteady procedure is to allow mesh adaptation prior to converging a steady state solution, not to obtain a time-accurate solution of an unsteady problem. Thus, an estimate of the error due to spatial discretization in the output of interest averaged over the current time step is developed. This error estimate is then used to drive an h-adaptation algorithm. Adaptation results demonstrate that the high-order discretizations are more efficient than the second-order method in terms of degrees of freedom required to achieve a desired error tolerance. Furthermore, using the unsteady adaptation process, adaptive RANS simulations may be started from extremely coarse meshes, significantly decreasing the mesh generation burden to the user.
Book Synopsis Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints by : Michael Hintermüller
Download or read book Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints written by Michael Hintermüller and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Provably Convergent Anisotropic Output-based Adaptation for Continuous Finite Element Discretizations by : Hugh Alexander Carson
Download or read book Provably Convergent Anisotropic Output-based Adaptation for Continuous Finite Element Discretizations written by Hugh Alexander Carson and published by . This book was released on 2020 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: The expansion of modern computing power has seen a commensurate rise in the reliance on numerical simulations for engineering and scientific purposes. Output error estimation combined with metric-based mesh adaptivity provides a powerful means of quantifiably controlling the error in these simulations, for output quantities of interest to engineers and scientists. The Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm, developed by Yano for Discontinuous Galerkin (DG) discretization, is a highly effective method of this class. This work begins with the extension of the MOESS algorithm to Continuous Galerkin (CG) discretization which requires fewer Degrees Of Freedom (DOF) on a given mesh compared to DG. The algorithm utilizes a vertex-based local error decomposition, and an edge-based local solve process in contrast to the element-centric construction of the original MOESS algorithm. Numerical results for linear problems in two and three dimensions demonstrate the improved DOF efficiency for CG compared to DG on adapted meshes. A proof of convergence for the new MOESS extension is then outlined, entailing the description of an abstract metric-conforming mesh generator. The framework of the proof is rooted in optimization, and its construction enables a proof of higher-order asymptotic rate of convergence irrespective of singularities. To the author’s knowledge, this is the first such proof for a Metric-based Adaptive Finite Element Method in the literature. A three dimensional Navier Stokes simulation of a delta wing is then used to compare the new formulation to the original MOESS algorithm. The required stabilization of the CG discretization is performed using a new stabilization technique: Variational Multi-Scale with Discontinuous sub-scales (VMSD). Numerical results confirm that VMSD adapted meshes require significantly fewer DOFs to achieve a given error level when compared to DG adapted meshes; these DOF savings are shown to translate into a reduction in overall CPU time and memory usage for a given accuracy
Book Synopsis An Adaptive Multiscale Finite Element Method by : Patrick Henning
Download or read book An Adaptive Multiscale Finite Element Method written by Patrick Henning and published by . This book was released on 2012 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Multiresolution Finite Volume Discretization of the Variational Multiscale Method by :
Download or read book Adaptive Multiresolution Finite Volume Discretization of the Variational Multiscale Method written by and published by . This book was released on 2011 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Computational Fluid Dynamics Review 2010 by : M. M. Hafez
Download or read book Computational Fluid Dynamics Review 2010 written by M. M. Hafez and published by World Scientific. This book was released on 2010 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains 25 review articles by experts which provide up-to-date information about the recent progress in computational fluid dynamics (CFD). Due to the multidisciplinary nature of CFD, it is difficult to keep up with all the important developments in related areas. CFD Review 2010 would therefore be useful to researchers by covering the state-of-the-art in this fast-developing field.
